Download Quantum Mechanics in 3 Dimensions

Quantum Mechanics in 3
Dimensions
Chapter 8
October 09
Modern Physics
Separation of time
Look for solutions harmonic in time.
Suppose the wavefunction has the form
October 09
Modern Physics
Free particle solutions
Look for solutions that factorize into a product of functions of
the independent variables:
Each term depends on a different variable so each must
be a constant.
October 09
Modern Physics
Free particle solutions
October 09
Modern Physics
Free particle solutions
For any (wave)vector k there is a solution
Any superposition of solutions is a solution so we
can find standing wave solutions of fixed energy.
Wavepackets, spherical waves… are constructed
by superposing solutions with different |k|, E
October 09
Modern Physics
Particle in a box
Consider a particle
confined to a 3
dimensional infinitely
deep potential well - a
“box”. Outside the
wave function
vanishes. Inside a
harmonic solution is a
product of standing
waves, each a linear
combination of
traveling waves.
October 09
Modern Physics
Particle in a box
October 09
Modern Physics
“Square box” or cube
If the sides have equal length
This special case has symmetry reflected in degeneracy
of energy levels.
October 09
Modern Physics
Ground state
The probability density in
a slice of space of constant z:
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Modern Physics
Excited states
For a cube, the the 121 state is a rotated 211 state.
October 09
Modern Physics
Degeneracy of states in a cube
For equal
length sides,
equal quantized
momentum and
energy can be
invested in the
three physically
equivalent
directions.
October 09
Modern Physics
Energy Levels
The energy level
spectrum is unlike the 1d case for which E~ n2
as different directions
have energy, and for
equal length sides each
level corresponds to
several wave states. For
unequal sides, the
degeneracy is broken
and levels separate.
October 09
Modern Physics
Waves along a tube
Example: A stretched box
has large L in one direction
so realtively small k and a
nearly continuous energy
spectrum is associated with
that direction.
October 09
Modern Physics
Leaky boxes
A box with penetrable walls
will have wave solutions
which leak out in three
dimensions.
October 09
Modern Physics
Applications
A generic ‘gas’ is a collection of non-interacting
particles. The properties derive from the spectrum
of wave states just described.
Examples:
Atomic or molecular gas in a macroscopic box
Conduction electrons in a metal
Nucleons inside a nucleus
In the last two examples, the particles are
individually strongly interacting but collectively the
effective potential is flat.
October 09
Modern Physics
Lessons
A box is a model for a bound wave in 3-d, a crude
model for an electron bound to a proton.
The solutions have features in common with
atomic electron bound states
- a spectrum increasing in density with energy,
- -degeneracy reflecting symmetry,
- complex pattern of nodes and lobes of probability
density
October 09
Modern Physics
Complex number review
October 09
Modern Physics
Spherical symmetry
We use Cartesian
coordinates x,y,z to describe
matter waves subject to
rectangular boundary
conditions.
Particle waves confined by a
spherical shell (eg, nucleons
in a nucleus) or spherically
symmetric central force are
most simply described with
spherical coordinates.
October 09
Modern Physics
Spherical coordinates
October 09
Modern Physics
Laplace operator in spherical
coordinates
Cartesian coordinates
Change variables:
It is a chore to prove this!
October 09
Modern Physics
Angular momentum operator
Isolate the angular dependence of the operator
The angular dependence resides in the quantity L2
which is (perhaps not surprisingly) related to orbital
angular momentum.
October 09
Modern Physics
Separation of spherical
variables
Write the wavefunction in terms of spherical coordinates
and suppose it factorizes into a product of functions:
Insert into the time independent Schrodinger
equation and isolate terms as we have before.
The separation is a bit trickier but the same idea.
October 09
Modern Physics
Separation of spherical
variables
Move r dependent terms to the left side.
The terms on the left and right depend on independent
variables so both must be constant. Call it -l(l+1).
October 09
Modern Physics
Separated equations
Radial equation
Angular equation
October 09
Modern Physics
Separation of angular equation
Separate polar and azimuthal angles
The 2nd term on the left that depends on azimuth must be
constant. Call the constant -m2.
Next: Solve in reverse order for
October 09
Modern Physics
Azimuthal equation
The equation for the azimuthal angle is familiar:
This azimuthal factor must be continuous as the
azimuthal angle runs from 0 to pi so m must be an
integer.
October 09
Modern Physics
Polar angle equation
This (Legendre) equation has
solutions for a given integer m
provided l is an integer. The
solutions are polynomials in cos
(theta) provided |m|< l+1.
October 09
Modern Physics
Spherical harmonics
The entire angular
wavefunction is
conventionally expressed as
a normalized “spherical
harmonic” function:
October 09
Modern Physics
Completeness
Fourier analysis:
Any function f(x) over x=[0,1] can be expressed
as a superposition of trig functions
Similarly, any function of spherical angles can be
represented as a superposition of spherical harmonics
October 09
Modern Physics
Angular momentum operator
A plane wave has a unique momentum.
Mathematically this is represented by the fact that
application of the momentum operator returns a
unique momentum value:
Similarly, a wave proportional to a spherical harmonic
has a unique orbital angular momentum magnitude and
z component:
October 09
Modern Physics
Angular momentum values
Angular momentum is quantized:
October 09
Modern Physics
Hydrogen Atom
Recall the separation of variables
Assuming the angular dependence
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Radial wave equation
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Effective potential
The first term is positive, an infinitely repulsive
force for nonvanishing l. Postive angular
momentum implies the particle can not be at the
origin!
The second term is attractive and dominates at
large r.
The net effective potential is a well in the radial
direction
October 09
Modern Physics
Effective potential
Put R = g/r
r
October 09
Modern Physics
Hydrogen atom radial
wavefunctions
The radial
equation has a
sequence of
solutions Rnl
labeled in order
of energy by
“principle
quantum number
n=1,2,… and
angular
momentum l with
l<n.
October 09
Modern Physics
Hydrogen atom complete
wavefunctions and energies
The complete time independent Schrodinger
equation and solution which are a radial factor Rnl
multiplies by an angular factor Ylm where
n=1,2,…
l = 1,2,…(n-1)
m = -l,…,+l
The energy of these states is independent of l and m
and given by Rydberg/Bohr formula!
October 09
Modern Physics
Spectroscopic notation
October 09
Modern Physics
Energy level diagram
The wave theory
tells us that the
energy levels are
(other than the
ground state) are
degenerate.
Optical transitions
connect states with
October 09
Modern Physics
Probability density
The probability an electron is in a volume dV is
For spherically
symmetric states, the
probability of an
electron within a shell
of radius r and
thickness dr is
October 09
Modern Physics
1s Radial Probability Distribution
The curve P1s(r)
representing the
probability of finding the
electron as a function of
distance from the
nucleus in a 1s
hydrogen-like state.
Note that the probability
takes its maximum
value when r equals a0/
Z.
October 09
Modern Physics
1s Cloud picture
The spherical
electron “cloud” for
a hydrogen-like 1s
state. The shading
at every point is
proportional to the
probability density
!! 1s(r) ! !2.
October 09
Modern Physics
S state probability distributions
For given n>0, nodes in the radial wavefunction
separate concentric “shells” of probability.
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Modern Physics
P state probability distributions
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Modern Physics
Higher excitations
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Modern Physics
Axial symmetry
(a) The probability density !! 211!2 for a hydrogen-like 2p state. Note
the axial symmetry about the z-axis. (b) and (c) The probability densities
!! (r) !2 for several other hydrogen-like states. The electron “cloud” is
axially symmetric about the z-axis for all the hydrogen-like states . !nlml(r)
October 09
Modern Physics
Cartesian p states
The 2p m=0 state is directional. Linear combinations
of m=+1 and m=-1 are similar, and associated with bonding.
October 09
Modern Physics
Exotic atoms
The hydrogen atom wave states apply to
similar matter atoms:
October 09
Modern Physics
Antihydrogen
Antihydrogen is detected through
its destruction in collisions with
matter particles. Pions (four light
colored dashed lines) point to the
annihilation point. Similarly, the
annihilation of the positron
produces a distinctive back-to-back
two-photon signature (two dashed
tracks at 180! to one another).
(Adapted from Nature, 419, 456–
459, October 3, 2002.)
October 09
Modern Physics
7.4: Magnetic Effects on Atomic Spectra—The
Normal Zeeman Effect
•  The Dutch physicist Pieter Zeeman showed the spectral
lines emitted by atoms in a magnetic field split into multiple
energy levels. It is called the Zeeman effect.
Anomalous Zeeman effect:
•  A spectral line is split into three lines.
•  Consider the atom to behave like a small magnet.
•  Think of an electron as an orbiting circular current loop of I =
dq / dt around the nucleus.
•  The current loop has a magnetic moment µ = IA and the
period T = 2πr / v.
• 
where L = mvr is the magnitude of the orbital
angular momentum.
The Normal Zeeman Effect
 
Since there is no magnetic field to
align them, point in random
directions. The dipole has a
potential energy
•  The angular momentum is aligned with the magnetic
moment, and the torque between and causes a
precession of .
Where µB = eħ / 2m is called a Bohr magneton.
• 
cannot align exactly in the z direction and
has only certain allowed quantized orientations.
The Normal Zeeman Effect
•  The potential energy is quantized due to the magnetic quantum
number mℓ.
•  When a magnetic field is applied, the 2p level of atomic hydrogen
is split into three different energy states with energy difference of
ΔE = µBB Δmℓ.
mℓ
Energy
1
E0 + µBB
0
E0
−1
E0 − µBB
The Normal Zeeman Effect
•  A transition from 2p to 1s.
The Normal Zeeman Effect
•  An atomic beam of particles in the ℓ = 1 state pass through a magnetic
field along the z direction.
• 
• 
•  The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the
mℓ = 0 state will be undeflected.
•  If the space quantization were due to the magnetic quantum number
mℓ, mℓ states is always odd (2ℓ + 1) and should have produced an odd
number of lines.
7.5: Intrinsic Spin
•  Samuel Goudsmit and George Uhlenbeck in Holland proposed that
the electron must have an intrinsic angular momentum and
therefore a magnetic moment.
•  Paul Ehrenfest showed that the surface of the spinning electron
should be moving faster than the speed of light!
•  In order to explain experimental data, Goudsmit and Uhlenbeck
proposed that the electron must have an intrinsic spin quantum
number s = ½.
Intrinsic Spin
•  The spinning electron reacts similarly to the orbiting electron in a
magnetic field.
•  We should try to find L, Lz, ℓ, and mℓ.
•  The magnetic spin quantum number ms has only two values,
ms = ±½.
The electron’s spin will be either “up” or
“down” and can never be spinning with its
magnetic moment µs exactly along the z axis.
The intrinsic spin angular momentum
vector
.
Intrinsic Spin
•  The magnetic moment is
•  The coefficient of
is −2µB as with
of relativity.
.
is a consequence of theory
•  The gyromagnetic ratio (ℓ or s).
•  gℓ = 1 and gs = 2, then
and
•  The z component of
•  In ℓ = 0 state
.
no splitting due to
.
there is space quantization due
to the intrinsic spin.
•  Apply mℓ and the potential energy becomes
7.6: Energy Levels and Electron Probabilities
•  For hydrogen, the energy level depends on the principle
quantum number n.
 
In ground state an atom cannot emit
radiation. It can absorb
electromagnetic radiation, or gain
energy through inelastic
bombardment by particles.
Selection Rules
•  We can use the wave functions to calculate transition
probabilities for the electron to change from one state to another.
Allowed transitions:
•  Electrons absorbing or emitting photons to change states when
Δℓ = ±1.
Forbidden transitions:
•  Other transitions possible but occur with much smaller
probabilities when Δℓ ≠ ±1.